Quantum information dynamics in a high-dimensional parity-time-symmetric system

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Quantum information dynamics in a high-dimensional parity-time-symmetric system

Zhihao Bian, Lei Xiao, Kunkun Wang, Franck Assogba Onanga, Frantisek Ruzicka, Wei Yi, Yogesh N. Joglekar, and Peng Xue

Phys. Rev. A 102, 030201(R) – Published 16 September 2020

Abstract

Non-Hermitian systems with parity-time ( $PT$ ) symmetry give rise to exceptional points (EPs) with exceptional properties that arise due to the coalescence of eigenvectors. Such systems have been extensively explored in the classical domain, where second- or higher-order EPs have been proposed or realized. In contrast, quantum information studies of $PT$ -symmetric systems have been confined to systems with a two-dimensional Hilbert space. Here, by using a single-photon interferometry setup, we simulate the quantum dynamics of a four-dimensional $PT$ -symmetric system across a fourth-order exceptional point. By tracking the coherent, nonunitary evolution of the density matrix of the system in $PT$ -symmetry unbroken and broken regions, we observe the entropy dynamics for both the entire system, and the gain and loss subsystems. Our setup is scalable to the higher-dimensional $PT$ -symmetric systems, and our results point towards the rich dynamics and critical properties.

Received 30 April 2020
Accepted 18 August 2020

DOI:https://doi.org/10.1103/PhysRevA.102.030201

Physics Subject Headings (PhySH)

PT-symmetric quantum mechanics Quantum foundations Quantum simulation

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Zhihao Bian^1,2, Lei Xiao¹, Kunkun Wang¹, Franck Assogba Onanga³, Frantisek Ruzicka^3,4, Wei Yi^5,6,*, Yogesh N. Joglekar ^3,†, and Peng Xue ^1,‡

¹Beijing Computational Science Research Center, Beijing 100084, China
²School of Science, Jiangnan University, Wuxi 214122, China
³Department of Physics, Indiana University Purdue University Indianapolis (IUPUI), Indianapolis, Indiana 46202, USA
⁴Institute of Nuclear Physics, Czech Academy of Sciences, Rez 250 68, Czech Republic
⁵CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
⁶CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei 230026, China

^*wyiz@ustc.edu.cn
^†yojoglek@iupui.edu
^‡gnep.eux@gmail.com

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Issue

Vol. 102, Iss. 3 — September 2020

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Images

Figure 1
Experimental setup. (a) Illustration of a four-mode $PT$ -symmetric qudit. (b) Schematic of the optical circuit used for simulating dynamics in the four-mode, passive $PT$ -symmetric system. Heralded single photons are generated via spontaneous parametric down-conversion and prepared in arbitrary qudit states using a polarizing beam splitter (PBS), wave plates with certain setting angles, and a beam displacer (BD). The coherent, lossy, nonunitary time evolution is realized by BDs, HWPs, and sandwich-type QWP-HWP-QWP setups, along with single-photon loss. For detection, projective measurements and quantum-state tomography are selected depending on the purpose, either of which is performed, both of which are realized by a PBS and (or) wave plates and a BD. Avalanche photodiodes (APDs) detect the signal and heralding photons.
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Figure 2
Scaled mode occupations of a $PT$ -symmetric system. (a) In the Hermitian limit with $γ = 0$ , mode occupation numbers are periodic with period $T = 2 π / J$ , and perfect state transfer occurs from mode $k$ to mode $5 - k$ at time $T / 2$ . (b) In the $PT$ -unbroken phase with $γ = 0.2 J$ , the occupation dynamics have period $T (γ)$ , and the total weights exceed unity indicating a non-trace-preserving time evolution. (c) At $γ = J$ , the scaled mode occupation grows as $t^{6}$ with time indicating that the exceptional point is of order four. (d) In the $PT$ -broken phase with $γ = 1.2 J$ , the scaled mode occupation grows exponentially with time. Time is measured in units of $J$ . Symbols: data; lines: Theory. Error bars are due to the statistical uncertainty and obtained based on assuming Poisson statistics. When not shown, error bars are smaller than the symbol size.
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Figure 3
Eigenvalues of the perturbed Hamiltonian. (a) The real and (b) imaginary parts of the eigenvalues of the perturbed Hamiltonian $H_{δ} = H_{PT} (γ = J) + i J δ | 1 〉 〈 1 |$ , measured in units of $J$ , scale as $δ^{1 / 4}$ , showing the enhanced sensitivity near the EP4. The initial states are eigenstates $| ψ_{δ} 〉$ of $H_{δ}$ . Experimental errors are calculated via Monte Carlo method; when not shown, error bars are smaller than the symbol size.
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Figure 4
Quantum information dynamics of the $PT$ -symmetric qudit. (a) Time-dependent entropy $S (t)$ for a qudit with a pure initial state $| ψ (0) 〉$ remains zero for any non-Hermiticity. (d) With a mixed initial state, $S (t)$ is constant when $γ = 0$ , and oscillates in the $PT$ -unbroken phase ( $γ = 0.2 J$ ) with period $T (γ)$ . At the EP4 ( $γ = J$ ) and in the broken $PT$ region, $S (t)$ reaches zero because the system approaches a pure state that is determined by the sole eigenstate at the EP or the mode with maximum amplification. In contrast, the subsystem entropies $S_{Gain} (t)$ and $S_{Loss} (t)$ show qualitatively similar behavior for (b), (c) pure and (e), (f) mixed initial qudit states. In each case, the entropies show oscillatory behavior for $γ < J$ and steady-state behavior for $γ \geq J$ . We restrict to $0 \leq t \leq 4.5$ due to the rapid growth of the scaled mode occupation at the EP4 and in the broken $PT$ region. Experimental errors are due to Monte Carlo method; when not shown, error bars are smaller than the symbol size.
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Figure 5
Reduced, instantaneously normalized density matrices (a)–(d) ${\tilde{ρ}}_{Gain} (t)$ and (e)–(h) ${\tilde{ρ}}_{Loss} (t)$ trace different paths inside the Bloch sphere. Experimental results are represented by colored symbols and their theoretical predictions are represented by dashed curves.
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